Sign-changing solutions for elliptic problems with singular gradient terms and \(L^{1}(\Omega )\) data

  • Stefano Buccheri


In this paper we deal with singular boundary value problems of the type
$$\begin{aligned} \qquad {\left\{ \begin{array}{ll} \displaystyle -\,\mathrm{div}\left( a(x,u)\nabla u\right) +b(x) \frac{|\nabla u|^{2} }{|u|^{\theta }}\text{ sign }(u) = f(x), &{} \text{ in } \Omega \text{, } \\ &{}\qquad \qquad \qquad \quad (0.1)\\ \qquad \qquad \qquad \quad \qquad \qquad \qquad \quad \;\, u = 0, &{} \text{ on } \partial \Omega \text{, } \end{array}\right. } \end{aligned}$$
where \(\Omega \) is a open bounded set of \(\mathbb {R}^N\) with \(N>2\), a(xt) is a Carathéodory function with polynomial growth with respect to t, b(x) is bounded and measurable, \(\theta \in (0,1)\) and f(x) belongs to \(L^{1}(\Omega )\). The main concern is to consider sign-changing solutions outside the energy space \(W_0^{1,2}(\Omega )\).


Quasilinear elliptic equations Singular gradient term Changing sign data 

Mathematics Subject Classification

35J62 35J75 


  1. 1.
    Arcoya, D., Martínez-Aparicio, P.J.: Quasilinear equations with natural growth. Rev. Mat. Iberoam. 24, 597–616 (2008)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arcoya, D., Barile, S., Martínez-Aparicio, P.J.: Singular quasilinear equations with quadratic growth in the gradient without sign condition. J. Math. Anal. Appl. 350, 401–408 (2009)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Arcoya, D., Carmona, J., Leonori, T., Martínez-Aparicio, P.J., Orsina, L., Petitta, F.: Existence and nonexistence of solutions for singular quadratic quasilinear equations. J. Diff. Eq. 249, 4006–4042 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Boccardo, L.: Dirichlet problems with singular and quadratic gradient lower order terms. ESAIM Control Optim. Calc. Var. 14, 411–426 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Boccardo, L., Croce, G.: Elliptic Partial Differential Equations: Existence and Regularity of Distributional Solutions. De Gruyter, Berlin (2014)MATHGoogle Scholar
  6. 6.
    Boccardo, L., Moreno, L., Orsina, L.: A class of quasilinear Dirichlet problems with unbounded coefficients and singular quadratic lower order terms. Milan J. Math. 83, 157–176 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Giachetti, D., Petitta, F., Segura de León, S.: Elliptic equations having a singular quadratic gradient term and changing sign datum. Commun. Pure and Appl. Anal. 11, 1875–1895 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Giachetti, D., Petitta, F., Segura de León, S.: A priori estimates for elliptic problems with a strongly singular gradient term and a general datum. Diff. Int. Eq. 26, 913–948 (2013)MathSciNetMATHGoogle Scholar
  9. 9.
    Leone, C., Porretta, A.: Entropy solutions for nonlinear elliptic equations in \(L^{1}(\Omega )\). Nonlinear Anal. 32, 325–334 (1998)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Porretta, A.: Some remarks on the regularity of solutions for a class of elliptic equations with measure data. Houston J. Math. 26, 183–213 (2000)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaSapienza Università di RomaRomeItaly

Personalised recommendations